I recently learned that there is a one-to-one correspondence between isomorphism classes of complex reductive groups and isomorphism classes of compact connected real Lie groups given by taking a compact real form in one direction and complexification in the other direction. I am happy with many parallels such as the existence of maximal tori, the definition of a Weyl groups, the classification of finite dimensional irreducible representations by dominant integral weights and so on. (Please correct me if I get any of the statements above wrong since I am just pulling them out from my memory.)
One question that bothers me is the following. In the complex reductive case, given a fixed maximal torus one can choose a Borel subgroup containing that torus, which is equivalent to choosing a collection of simple roots in the root system; is there an analog of this in the compact connected real Lie group case? (I don't know how to properly define a root system for a compact connected real Lie group without passing to the corresponding complex reductive group either.) Thanks a lot.